with \(\zeta =0.8\)  for \({R_{\textnormal {acc}}}=5R_{\mathrm {p}}\)  and \(\zeta \rightarrow 1\)  when \({R_{\textnormal {acc}}}\gg r\) . For a given accretion rate, the density depends on the filling factor but the velocity does not. These are the classical formulae for an accretion flow (e.g. [1]}, [2]}) and have been verified to apply to planets by [3]}, [4]} through one-dimensional radiation-hydrodynamical simulations. For reference, the shock Mach number is in the range \(\mathcal {M}\approx 10\) –40 [4]}, and the ram pressure of the incoming gas is given by (e.g. [6]})
\(P_{\textnormal {ram}}=&~ \rho (R_{\mathrm {p}}) v(R_{\mathrm {p}})^2 = \frac{\dot{M} v(R_{\mathrm {p}})}{4\pi R_{\mathrm {p}}^2f_{\mathrm {fill}}}\\\approx &~171~\mathrm {erg\,cm}^{-3}\frac{1}{f_{\mathrm {fill}}} \left(\frac{\dot{M}}{M_{\mathrm {J}}\,\mathrm {yr}^{-1}}\right) \\ &\left(\frac{M_{\mathrm {p}}}{3~M_{\mathrm {J}}}\right)^{1/2}\left(\frac{R_{\mathrm {p}}}{2~R_{\mathrm {J}}}\right)^{-5/2}\zeta ^{1/2}.\) 
